# An Introduction to Algebraic Topology

Springer Science & Business Media, Jul 22, 1998 - Mathematics - 438 pages
There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces.

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My go to Alg.Top book. Fantastic I recomend to everyone.

### Contents

 CHAPTER 0 1 Brouwer Fixed Point Theorem 2 Categories and Functors 6 CHAPTER 1 14 Convexity Contractibility and Cones 18 Paths and Path Connectedness 24 CHAPTER 2 31 Affine Maps 38
 Homology and Attaching Cells 189 CW Complexes 196 Cellular Homology 212 EilenbergSteenrod Axioms 230 Chain Equivalences 233 Acyclic Models 237 Lefschetz Fixed Point Theorem 247 Tensor Products 253

 CHAPTER 3 39 The Functor n₁ 44 n₁Są 50 Free Abelian Groups 59 The Singular Complex and Homology Functors 62 Dimension Axiom and Compact Supports 68 The Homotopy Axiom 72 The Hurewicz Theorem 80 Exact Homology Sequences 93 Reduced Homology 102 Homology of Spheres and Some Applications 109 Barycentric Subdivision and the Proof of Excision 111 More Applications to Euclidean Space 119 Simplicial Approximation 136 Abstract Simplicial Complexes 140 Simplicial Homology 142 Comparison with Singular Homology 147 Calculations 155 Fundamental Groups of Polyhedra 164 The Seifertvan Kampen Theorem 173 Attaching Cells 184
 Universal Coefficients 256 EilenbergZilber Theorem and the Kunneth Formula 265 Basic Properties 273 Covering Transformations 284 Existence 295 Orbit Spaces 306 Group Objects and Cogroup Objects 314 Loop Space and Suspension 323 Homotopy Groups 334 Exact Sequences 344 Fibrations 355 A Glimpse Ahead 368 Cohomology Groups 377 Universal Coefficients Theorems for Cohomology 383 Cohomology Rings 390 Computations and Applications 402 Bibliography 419 Notation 423 Index 425 Copyright